arXiv:2611.13904v1 Announce Type: cross Abstract: Let $\mathcal{M}$ and $\mathcal{M}_\sigma$ denote, respectively, the sets of Borel and $\sigma$-invariant probabilities on the uncountable set $\Omega=\{1,2,...,d\}^\mathbb{N}$. Given a H\"older Jacobian $J$, the dual Ruelle operator $\mu \in \mathcal{M} \mapsto \mathcal{L}_{J}^* (\mu)$ will play the role of a dynamical channel; generalizes the action of a stochastic matrix on probability vectors. Since $\mathcal{L}_{J}^*$ generally breaks $\sigma$-invariance, we introduce the \emph{ni-entropy} $h_{ni}(\mu)$ for non-invariant measures, employing the concept of the expected value function on cylinders of size one. $h_{ni}(\mu)$ differs from Shannon entropy but extends the Kolmogorov-Sinai entropy from $\mathcal{M}_\sigma$ to $\mathcal{M}$; $h_{ni}$ is concave and u.s.c. This leads us to the concept of ni-pressure $\mathcal{P}(A)$ of a potential $A:\Omega \to \mathbb{R}$. $\mu\in \mathcal{M}$ is not a density with respect to a dominating measure. We define an {\it ni-Kullback-Leibler divergence} $D_{ni}(\rho || \nu)$ not assuming absolute continuity. We investigate concepts of Information Theory such as Minimal Output Entropy, Donsker-Varadhan variational principle, a variational formula for $h_{ni}$, bounds for divergence, information projections, and their connections to the ni-pressure $\mathcal{P}$.